International Journal of Heat and Mass Transfer 52 (2009) 564–573

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Gold nanoshell density variation with laser power for induced hyperthermia

Jerry Vera, Yildiz Bayazitoglu *

Department of Mechanical Engineering and Materials Science, Rice University, 6100 Main Street, MS 321, Houston, TX 77005-1892, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 May 2008Received in revised form 23 June 2008Available online 25 August 2008

Keywords:Photothermal therapyHyperthermiaLasersNanoshellCancer therapyTissue optics

0017-9310/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2008.06.036

* Corresponding author. Tel.: +1 713 348 6291.E-mail address: bayaz@rice.edu (Y. Bayazitoglu).

The thermal profile effects of nanoshell density, laser power, and laser arrangement are presented forideal cases of nanoshell-assisted photothermal therapy. A one-dimensional thermal model utilizing theP1 approximation is used to simulate the penetration of laser radiation and subsequent heating of1-cm slabs of nanoshell-embedded tissue exposed to a 633-nm collimated light source. It is shown thatadding too many nanoshells or increasing power can cause overheating in the entry region while leavingthe rear region heated only by conduction, producing an undesirable temperature differential. An oppos-ing dual-laser approach is presented that mitigates this issue.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Over the past several years, much interest has been generatedsurrounding the use of light-absorbing gold nanoshells as a newtool to fight cancer using photothermal tumor therapy. It has beenrepeatedly shown that by varying the relative dimensions of thecores and thickness of the shell in such nanoparticles, the opticalresonance can be precisely and systematically varied over a broadregion ranging from the near-UV to the mid-infrared. This regionincludes the near-infrared (NIR) wavelength range where tissuetransmissivity peaks [1–3]. By tuning gold nanoshells to maximizeplasmon resonance, in this region these particles can efficientlyabsorb incident radiation and generate heat, which in turn raisesthe temperature of their immediate surroundings. Using any of avariety of means of injecting gold nanoshells into biological tissue,one can increase the temperature of the tissue by exposing a nano-shell-embedded tissue ‘‘composite” to laser light via minimallyinvasive fiber optics, and with increased absorption focused inthose regions containing nanoshells, hyperthermia can be achievedin target areas with minimal consequence to non-nanoshell-embedded tissue [4]. Specifically, this procedure can be used toproduce hyperthermia in malignant tumors with the goal of locallyinducing cellular necrosis or weakening tumorous tissue into astate where necrosis can occur using a lower required dose ofionizing radiation from radiation therapy [5]. Because gold isconsidered a biocompatible material [3], this procedure is thus

ll rights reserved.

far considered safe, and both methods of inducing necrosis arefavorable to traditional radiation therapy, where the DNA of allcells in the path of the radiation beam are subject to damage.

The purpose of this study is to investigate the feasibility of goldnanoshell-assisted photothermal therapy by focusing on themechanical aspects of the problem: namely radiative transport,distributed heating, and thermal conduction. A simulation of opti-cal diffusion based on radiative transport theory using the P1

approximation in a solid medium with homogenously distributedgold nanoshells is used and applied to biological tissue as the basemedium. Using a code developed by Bayazitoglu et al., severalcases are presented of laser diffusion through and the subsequentheating of biological tissue infused with various gold nanoshell dis-tribution densities. This study also simulates the heat distributionas a result of varying input power levels and reports the resultingtemperature distributions at subsequent gradations of time. Thegoal is to present key examples of those parameters that wouldbe required to raise a tissue to a given elevated temperature char-acteristic of hyperthermia while maintaining that the affected tis-sue is warmed evenly and steadily across its depth. Also noted isthe time progression of this increase in temperature so that meth-ods of raising and holding a tissue to a given hyperthermic state fora given time duration can be devised and utilized by therapists inthe field of oncology.

2. Analysis

Previous studies have shown that a semitransparent mediumcan be engineered by embedding gold nanoshells in a transparent

Nomenclature

SymbolsB’s analytical constantsBi Biot numberC’s analytical constantsc specific heat (J/kg K)Fo Fourier numberG irradiance (W/m2)h convective heat transfer coefficient (W/m2 K)I intensity of radiationk thermal conductivity (W/m K)L length of slabM number of nodesNT number of particles per unit volume or particle concen-

tration (particles/m3)Q efficiencyq heat flux (W/m2)ri nanoshell core radiusro nanoshell shell radiusT temperature (�C or K)DT difference between highest and lowest temperature in

target tissue regiont time (s)Dt time increment (s)u000 heat generation (W/m3)z axis of coordinate system inside medium pointing

towards the depth normal to boundary (m)Dz spatial increment (m)

Greek symbolsa thermal diffusivity (m2/s)

b extinction coefficient (m�1)j absorption coefficient (m�1)lc coshh laser angle of incidenceq density (kg/m3)q reflectivityrs scattering coefficient (m�1)s optical depth or optical locationsL optical lengthx scattering albedon1 analytical constant

Subscriptsc correspond to collimated radiationd correspond to diffuse radiationin correspond to inputmd correspond to host mediumR correspond to total radiation, summation of diffuse and

collimated radiations correspond to nanoshellstot correspond to total properties for nanoshell/host med-

ium composite

Greek subscriptsb correspond to extinctionj correspond to absorptionk spectralr correspond to scattering

J. Vera, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 52 (2009) 564–573 565

slab, and altering the number of nanoshells spread homoge-neously throughout this slab can vary the levels of attenuationof light penetrating through this medium, which in turn variesthe amount of absorption and heating made in the compositealong its depth [6]. When light enters a nanoshell-embeddedslab, propagating light is absorbed and scattered by both thehost medium and the embedded nanoshells. The extent ofabsorption and scattering made by these two constituents canbe analyzed linearly, since radiative transfer theory showsradiative intensity of a propagating light through an absorbingand isotropically scattering, one-dimensional plane parallelmedium with constant radiative properties is a linear function[7]. Cases where nanoshell density is high, and inter-nanoshelldistance is within a few particle diameters of one another arenot considered here, since induced dipole–dipole interactionsbetween neighboring nanoparticles will induce non-linear near-field effects [8].

2.1. Problem description

In this study, a collimated light source illuminates a 1-cm slabof human biological tissue with varying degrees of nanoshell den-sities homogeneously distributed within. In a clinical setting, thenanoshells are assumed to be added either by direct injection ontothe site of interest, or intravenously with eventual deposition intothe desired region due to vasculature abnormalities in tumoroustissue [9]. This slab represents what would be a pea-sized tumorthat has developed within a healthy tissue, and it is surroundedon both sides by regions of healthy tissue. These regions providea thermal conduction buffer zone which acts as a limited heat sink,without which temperatures in the illuminated tissue would esca-

late to higher than accurate levels. These are 1-cm long each andare located to the left and right of the nanoshell-embedded tissue.

The heat source assumed here is a 633-nm helium-neon laserdelivered by a fiber optic cable that is inserted into the body in aminimally invasive procedure, is applied directly onto the site ofa tumor, and projects into the direction of its depth. This projectionis located on one end of the slab in the first round of studies andthen on the two opposing ends of the slab in the second round ofstudies (see Fig. 1). A wavelength of 633 nm was chosen sinceabsorption due to whole blood and water is low at this value, thusallowing for high transmissivity [10]. It is not surprising that as aresult, the vast majority of tissue properties readily found in the lit-erature were listed at this wavelength or at 1035 nm whereabsorption by these constituents is at a minimum [11].

Several types of tissue are examined including human brain,breast, subcutaneous fat, liver, and skin. The optical and thermalproperties of these tissues are pertinent to the penetration of lightand the subsequent heating and spread of heat throughout the slabdepth. The tissue properties utilized are listed in Table 1. Theywere recorded at room temperature, and for the purpose of sim-plicity in the current analysis, they are assumed to remain constantas the temperature changes.

Nanoshells are added to the matrix in each of the tissues abovewith varying particle distribution densities, NT, varying between 0,7 � 1014, 7 � 1015, and 7 � 1016 particles per cubic meter. Studyingthe tissue optical properties listed in Table 1, we can see that in allcases, the absorption per unit length is relatively low in compari-son to the scattering. As a result, the goal of the addition of thenanoshells to the matrix is to augment the amount of absorptionthat occurs in the tissue while keeping any additional scatteringto a minimum. To accomplish this, a suitable nanoshell geometry

Fig. 1. One-dimensional nanoshell-embedded tissue thermal analysis setup.

Table 1Utilized tissue thermal and optical properties at 633 nm

Tissue Temperature(�C)

Index ofrefraction

Conductivity(W/m K)

Specific heat(J/kg K)

Mass density(kg/m3)

Abs (m�1)at 633 nm

Sca (m�1)at 633 nm

Brain, human 37 1.37 0.515 3600 1035 26 5700Breast, human 37 1.44 0.499 3920 990 20 39,400Fat, subcutaneous 37 1.44 0.25 3920 916 10 6690Liver, human 37 1.367 0.497 3600 1050 320 41,100Skin, dermis 37 1.55 0.293 3150 1100 180 39,400

Estimates and substitutions used where needed. Source: Refs. [10–12].

566 J. Vera, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 52 (2009) 564–573

is selected to target peak absorption and exhibit minimal scatter-ing when exposed to a 633-nm excitation.

The extinction efficiency spectra of nanoshells can be calcu-lated quite accurately using Mie theory [13]. After modeling sev-eral combinations of nanoshell core sizes and shell thicknesses, itwas found that a nanoshell with a core radius of 16 nm and shellthickness of 5 nm produces desirable results. The absorption andscattering efficiency spectra of such a shell is shown in Fig. 2. Thisspectra is based on a Mie scattering approximation of an array ofgold nanoshells surrounded by water with a two sigma Gaussiandistribution between ±2 nm for r1 and ±0.5 nm for the shell thick-ness. The variation in shell thickness and core size is used to fac-

400 450 500 550 600

1

2

3

4

5

6

7

Wavelen

Effi

cien

cy

scaabsext

ro = 21 nm

ri = 16 nm

Effciency Spectrum of ri = 16 nm, ro

Fig. 2. Scattering and absorption spectra of [ri,ro] = [16 nm,21 nm] nanoshell suspensionshell population.

tor in for a likelihood of polydispersity in the nanoshellsuspension, and the inclusion of a surrounding medium is neces-sary to form the boundary conditions used in the Miecalculations. Water was chosen because it was deemed theclosest approximation to tissue, which has a high water content,however, it should be noted that the resulting extinction spectracalculated by this method does not include absorption linescaused by water.

The nanoparticle spectral absorption and scattering cross-sec-tions jsk and rsk, respectively, are deduced from the spectralabsorption and scattering efficiencies of the nanoshells Qrk andQjk under the following relations:

0 650 700 750 800

gth (nm)

637 nm , 0.92068

631 nm , 5.9492

= 21 nm gold nanoshell suspension

from Mie theory approximation with line broadening due to likely polydispersity in

J. Vera, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 52 (2009) 564–573 567

rsk ¼ pr2oQrkNT ð1Þ

jsk ¼ pr2oQjkNT ð2Þ

again, where NT is the number of nanoshells per unit volume. Theabsorption and scattering cross-sections in Eqs. (1) and (2) definethe degree of absorption or scattering performed on the incominglight per unit depth at a particular wavelength by the nanoshellarray. Increasing the density of nanoshells in the array increasesthe respective absorption and scattering. Increasing the nanoparti-cle cross-sectional area will have the same effect, however in thepresent study, nanoparticle efficiency is emphasized, therefore theoptical properties and the geometry detailed in Fig. 2 is heldconstant, and only changes in distribution density are considered.

As noted, the tissue optical properties listed in Table 2 denotethe respective absorption and scattering of the tissue alone at633 nm. To model the combined extinction caused from the syn-thesis of both media, we apply a superposition principal statingthat the combined extinction from the nanoshell-embedded tissuemedium is equal to the sum of the extinctions from its constituentparts. The extinction cross-section is defined as the sum of theabsorption and scattering cross-sections of a medium [7]. Herewe have contributions to extinction from the nanoshell array andthe host tissue in which they are dispersed. When broken downinto their absorptive and scattering components, we have thefollowing:

btot�k ¼bsk þ bmd�k

btot�k ¼jsk þ rsk þ jmd�k þ rmd�k

btot�k ¼pr2oQjkNT þ pr2

oQrkNT þ jmd�k þ rmd�k ð3a;b; cÞ

The relative distinction between scattering and absorption isexpressed in the scattering albedo for the composite system:

xtot�k ¼rtot�k

btot�k

ð4Þ

where the spectral scattering coefficient for the total composite sys-tem can be expressed as

rtot�k ¼ rs�k þ rmd�k ð5Þ

we have the following relation for the total composite scatteringalbedo:

xtot�k ¼rsk þ rmd�k

pr2oQjkNT þ pr2

oQrkNT þ jmd�k þ rmd�kð6Þ

The scattering albedo here is used in the P1 approximation shown inthe following section to approximate the resulting collimated anddiffuse incident radiation and heat flux as functions of optical depthinto the medium. The heat flux profile is subsequently used togenerate the time-dependent temperature profile.

Table 2Analytical constants

B1 � 1lc

xkð1� qkð0ÞÞqin;k1lc� n2

1

B3 2þ �k

ð2� �kÞð1�xkÞn1

B5 � �k

2� �k

xk

1�xkð1� qkð0ÞÞqin;k þ

4ð1� �kÞ2� �k

Hc0;k

B7 �2en1sLkþ

�k

ð2� �kÞð1�xkÞn1e�n1�sLk

B9 � �k

2� �k

xk

1�xkð1� qkð0ÞÞqin;ke

�sLklc þ 4ð1� �kÞ

2� �kHcL;k

2.2. Radiative analysis

To model the diffusion of light intensity, the solution to the nearinfrared heating of a one-dimensional, conducting and radiativelyparticipating medium as presented in [6] is utilized. The assump-tions taken here are nearly identical in that: (1) the medium isone-dimensional, (2) emission from the medium and nanoshellsis negligible, (3) the host medium is homogeneous, (4) the goldnanoshells in the medium are distributed evenly, and (5) all ofthe nanoshells in the medium have the same size and aspect ratio(core diameter/shell diameter). This study differs where the hostmedium is assumed to be transparent to spectral irradiation, sincethe tissue media utilized here possess inherent absorption andscattering.

This solution uses the P1 approximation described by Modestin [7]. As shown in [6], it manifests itself as a relation betweenthe diffuse intensity of radiation with diffuse irradiance G(s) anddiffuse radiative heat flux q(s) as functions of optical depth. Thesolution for the radiative heat flux profile is borrowed and thenused in the FTDT analysis described in the next section to as avolumetric heating term that varies along the depth of themedium.

2.3. FDTD formulation

A finite difference time domain (FDTD) calculation is performedon the entire length of the tissue calculating the local temperatureevolution over time. As stated, it is assumed that a fiber optic cableis inserted into the body during a minimally invasive procedure,and laser power is applied directly onto the site of a tumor, withentry of collimated radiation on one end of the slab in the firstround of studies, then entry from the two opposing ends of the slabin the second round of studies.

Five assumptions are made in solving the temperature distribu-tion: (1) the physical and thermal properties of the host mediumare constant, (2) the nanoshells contribute negligible additionaleffects to the physical and thermal properties of the medium, (3)the medium is considered homogeneous, (4) the medium isone-dimensional, and (5) emission is negligible. The energyequation incorporating these assumptions can be expressed as:

1amd

dTdt¼ d2T

dz2

u000

kmdð7Þ

where u000 is the local heat generation spectrum across the depth ofthe slab created by the absorption of incident radiation into thecomposite. It is defined as the negative sign of the divergence ofradiative heat flux, which is taken from the radiative heat transferanalysis.

B2 2� �k

ð2� �kÞð1�xkÞn1

B4 � 2þ �k

ð2� �kÞð1�xkÞ1lc

� �

B6 � 2en1sLkþ

�k

ð2� �kÞð1�xkÞn1en1sLk

0B@

1CA

B8 �2e1sLk=lc � �kð2��kÞð1�xkÞ

e�sLk=lc

lc

n1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1�xkÞ

p

Table 3Values utilized in FDTD formulation

Parameter Value

L 0.01 mM 83Dz 0.00012048 mDt 0.05 s

568 J. Vera, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 52 (2009) 564–573

u000 ¼ �dqRk

dzð8Þ

The local heat transfer spectrum is treated as an array of localizedpoints of heat generation that raises the temperature of the imme-diate surroundings. The spectral radiative heat flux, qRk(s), isderived from combining components from the heat generationcaused by collimated radiation qck(s) and the heat generationcaused by diffuse radiation qdk:

qRkðsÞ ¼ qckðsÞ þ qdkðsÞ ð9Þ

The collimated component, is derived in as a function of inputpower, reflectivity, and incoming direction:

qckðsÞ ¼ ð1� qÞqine�slc ð10Þ

where the componential fraction due to the angle of incidence is

lc ¼ cos h ð11Þ

In all cases presented here, the laser penetrates the tissue in a direc-tion normal to the entry boundary, i.e. h = 0, therefore

qckðsÞ ¼ ð1� qÞqine�s ð12Þ

The diffuse component is given as

qdkðsÞ ¼ C1ef1s þ C2e�f1s þ B1e�s ð13Þ

where C1 and C2 are analytical constants obtained by solving:

C1

C2

� �¼

B2 B3

B6 B7

� ��1 B4B1 B5

B8B1 B9

� �ð14Þ

The B’s and n1 constants are tabulated in Table 2.For brevity, the scattering albedo in Eq. (6), xtot�k, is presented

as xk. The full derivation of Eq. (13) can be found in reference [6].The final result is an equation for the spectral radiative heat fluxqRk(s) as a function of the optical depth s. The optical depth is afunction dependent on the total extinction coefficient btot�k asshown in Eq. (3-b), and the current location z along the one-dimen-sional axis into the slab:

sk ¼ btot�kz ð15Þ

The total optical depth skL is

skL ¼ btot�kL ð16Þ

In MATLAB, qRk(s) is expressed as an array of M nodes across theirradiated slab of total length L. The discretized length betweeneach node is labeled as Dz and is calculated as

Dz ¼ LM � 1

ð17Þ

Table 4Governing finite-difference equations for respective nodes

Nodes Finite difference equation

1 Ttþ1m ¼ Tt

m þ 2BiFo T1 � Ttm

� �þ 2Fo Tt

mþ1 � Ttm

� �2–81 Ttþ1

m ¼ Ttm þ Fo Tt

mþ1 þ Ttm�1 � 2Tt

m

� �82–164 Ttþ1

m ¼ Ttm þ Fo Tt

mþ1 þ Ttm�1 � 2Tt

m

� �� FoðDzÞ2

kmdbtotal�ku000fd

165–249 Ttþ1m ¼ Tt

m þ Fo Ttmþ1 þ Tt

m�1 � 2Ttm

� �250 Ttþ1

m ¼ Ttm þ 2BiFo T1 � Tt

m

� �þ 2Fo Tt

m�1 � Ttm

� �

where L is measured in meters. Table 3 outlines the values used andcalculated here.

Thus qRk(s) is expressed as qRk(z)i in discretized space where z isthe direction and i is the number of steps along the nodes of thatdirection indicating the length. The discretized spectral radiativeheat flux function obtained here is then used in Eq. (8) and incor-porated into the unsteady heat transfer equation in (7) to simulatethe effects of distributed heating by a laser source entering at z = 0and model the change in temperature. For the cases where thereare two laser sources, one at z = 0 and the other at z = L, a symme-try and linear addition theorem is applied. A laser entering fromthe right side of homogeneous tissue is assumed to have the sameoptical effects as one entering from the left side. Thus, the resultsfrom the heat flux calculations from one side can be applied tothe other. We apply this principle to u000, Eq. (8), the negative deriv-ative of the heat flux function. The MATLAB function fliplr isused to create a new array uR

000(z) from z = 0 to z = L that is essen-tially the same array with values counting backwards from z = Lto z = 0. The original array in (8) is renamed uL

000(z), and the summa-tion relation is applied to form a new array, utot

000(z):

u000totðzÞ ¼ u000L ðzÞ þ u000R ðzÞ ð18Þ

The total heat generation spectrum in (18) conducts heat through-out the tissue and applied to (7), where the thermal diffusivity amd

of the medium is defined by:

amd ¼kmd

qmdcmdð19Þ

For the FTDT analysis, the Fourier number of the system must bebelow the critical value of Focrit = 0.50. The Fourier number isdefined by:

Fo ¼ amdDtDz ð20Þ

The boundaries of this system are additional 0.01-m slabs oneach side of the slab system exposed to laser irradiation. Heatis allowed to spread into this system as the center slab is irra-diated. The boundary regions are assumed represent surround-ing tissue around the tumor site in the body, and for thepurposes of this analysis, they are assumed to be composedof the same material as the slab of interest, so the thermalproperties are identical. However, the boundary regions arenot exposed to radiation, and thus do not exhibit any localizedheat generation.

At the ends of the two boundary regions, for simplicity, the codeassumes a complete cutoff of the computational space. This is sim-ulated using a convection model on the first and final nodes of thesystem where the Biot number is set equal to 0 due to a convectiveheat transfer coefficient, h, of 0.

Bi ¼hDZkmd

ð21Þ

As a result of the statements above, the FDTD model utilizing thediscretized mode of (8) is divided into five computational regions:the left cutoff node, the left boundary region, the medium of inter-est region (the nanoshell composite slab), the right boundary re-gion, and the right cutoff node. The equations used to modelthese regions are delineated in Table 4.

Region Nodal length Length

Left Cutoff Node 1 PointLeft Buffer Zone 79 0.01 mIrradiated Slab Region 82 0.01 mRight Buffer Zone 84 0.01 mRight Cutoff Node 1 Point

J. Vera, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 52 (2009) 564–573 569

3. Results and discussion

Resulting time-dependent temperature profiles were preparedfor all tissues presented using varying particle distribution densi-ties, NT, of nanoshells varying between 0, 7 � 1014, 7 � 1015, and7 � 1016 particles per cubic meter. Using these densities and thenanoshell geometry and optical properties shown in Fig. 2, themaximum possible augmentation to the inherent scattering andabsorption of the host tissues (Table 1) is shown by the tabulationin Table 5.

For every density variation, four power levels for the input laserbeam were used: 5000 W/m2, 10,000 W/m2, 15,000 W/m2, and20,000 W/m2. This process was done using a single laser from theleft side at z = 0, and was repeated using a dual laser configurationat z = 0 and z = L, both facing each other, pointing into the nano-shell-embedded tissue.

The total results are too numerous to include here, so only themost significant findings are presented which showed the generaltrends found between all cases. In general, the change of the tem-perature profile were very similar between five major categories ofvariation. They are:

(1) increasing the nanoshell density distribution while holdingpower intensity constant,

(2) increasing the power intensity while holding the nanoshelldensity constant

(3) using two (dual) facing lasers as opposed to one single laser,(4) using the dual-laser configuration and holding the nanoshell

dispersion density constant and increasing the power, and(5) using the dual-laser configuration and increasing the nano-

shell density distribution while holding the power intensityconstant.

a

Fig. 3. Effect of increased nanoshell distribution density on induced hyperthermia foridentical tissue with 7 � 1016 nanoshells/m3.

Table 5Contribution to scattering and absorption by [ri, ro] = [16, 21] nm nanoshells

Dispersion density(m�3)

Nanoshell area(m2)

Absorption efficiency(m�1)

Scattering(m�1)

0 1.38544E�15 5.9492 0.929687.00E+14 1.38544E�15 5.9492 0.929687.00E+15 1.38544E�15 5.9492 0.929687.00E+16 1.38544E�15 5.9492 0.92968

Fig. 3 shows the effect of nanoshell density variation on a0.01 cm slab of breast tissue. The graph is divided into three parts.The center shows the evolution of the temperature profile overtime in the nanoshell-embedded region in increments of 5 min.This region represents the section of tissue heated by the laser.To the left and right of this section are the bordering tissue regionswhich represent the surrounding healthy tissue which warm byconduction.

Two metrics are used to quantify the change in heating of thetemperature profile. The first is the time taken for the entry node(z = 0) to reach 55 �C. Fifty-five degree celcius was chosen since itis a higher temperature than most cases presented in the literaturefor hyperthermia, and it is considered an extreme upper bound forthe cases presented. The second metric is the temperature differ-ence between the highest and lowest temperatures induced inthe central tissue region at the point in time where the entry notereaches 55 �C and the simulation stops. Since the general goal of in-duced hyperthermia is to raise the temperature of a desired regionof tissue, it is advantageous for the elevated temperature distribu-tion to be as even as possible. Thus, a high DT in the 1-cm targetedtissue region is undesirable. Efforts should be made to minimizethis number. Furthermore, it is also desirable to leave the healthysurrounding tissue largely unaffected by the propagation of heatfrom the laser beam. Thus cases where the target region tempera-ture goals are met while leaving temperatures in the left and right‘‘buffer” zones relatively close to normal body temperature(36.8 �C) are preferred.

In Fig. 3, we see that the control tissue in (a) requires 50.3 minfor the entry node to reach 55 �C and has a DT of 5.11 �C across thetissue in the region from 0.01 m to 0.02 m. It can be seen from thedifferent profile lines at 5 min, 10 min, etc. that the DT slowly in-creases as time progresses, so if a desired temperature increase is

b

breast tissue at 10,000 W/m2: (a) control tissue with zero nanoshell addition, (b)

efficiency Shell absorption coefficient(m�1)

Shell scattering coefficient(m�1)

0 0.0005.770 0.902

57.696 9.016576.959 90.161

570 J. Vera, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 52 (2009) 564–573

only 45 �C, then it only requires between 15 and 20 min of heatingtime and the DT would be between 4.28 to 4.58 �C. One should alsonote that without nanoshells, the total absorption cross section inthe targeted tissue region is only 20 m�1. This explains the highheating time. The longer it takes for the tissue to heat to thedesired temperature, the more time that is allowed for heat topropagate to the surrounding tissue via conduction. This is shownin (a), as the temperature 0.01 m to the left of central tissue regionis 53 �C at 50.3 min, and the point 0.01 m to the right of the centralregion is as high as nearly 48 �C. Such broad heating is undesirablefor this study, since we wish only to target the central region.

Raising the nanoshell content to 7 � 1016 particles per meter,shown in Fig. 3b, increases the total absorption coefficientnearly 30 times to 594.13 m�1. This lowers the required heatingtime to 8.08 min, which is desirable, but it also increases thetemperature differential across the 0.01-m central region to13.7 �C.

Fig. 4 shows the effects of altering the laser power intensitywhile holding nanoshell distribution density constant. The tissuehere is the dermis layer of human skin, with an inherent absorp-tion coefficient of 180 m�1 and a scattering coefficient of39,400 m�1. The presence of nanoshells raises the absorption to185 m�1 and the scattering to 39,401 m�1. We can see that in thecontrol case at 5000 W/m2, shown in (a), 28.3 min is required toraise the tissue to 55 �C and the maxDT is 10.1 �C. Raising the laserpower while holding the nanoshell density constant lowers therequired heating time to 9.28 min but increases the DT to14.6 �C. The decrease in heating time is beneficial, since thewarming of surrounding tissue is lowered, however the uneventemperature distribution results in a lack of heating to the rearregion of the targeted central zone, where the temperature onlyreaches 41 �C.

Comparing Figs. 3b and 4b, it can been seen that despitedifferences in density, specific heat, and conductivity betweenbreast tissue and skin dermis, the effects of adding nanoshells orincreasing power density are very similar: both lower heating timewhile increasing DT. For the purposes of evenly heating a long sec-tion of tissue, both methods yield undesirable results.

When using two opposing lasers to heat the selected tissue re-gion, however, the results improve. Fig. 5 shows the effect of alter-ing the laser configuration while holding all other variables

a b

Fig. 4. Effect of increased power on induced hyperthermia for dermal skin tissue withidentical tissue exposed to 10,000 W/m2 laser.

constant, using subcutaneous fatty tissue with 7 � 1014 nano-shells/m3 density and a power level of 15,000 W/m2. In (a), thecontrol tissue requires 7.55 min for the entry point to reach55 �C. The offset peak from the entry node to slightly inside thedepth of the tissue is characteristic of the deeper penetration ofincident radiation allowed by tissues of lower scattering, sincethe total scattering coefficient here is 6690.9 m�1 compared to�39,000 m�1 as in the previous examples. Switching to two oppos-ing lasers, shown in (b) lowers the required heating time to 5.5 minand lowers the maximum DT to 4.96 �C. In addition, the tempera-ture drops off significantly in the surrounding tissue, making thiscombination highly attractive for the targeted heating of only thecentral region.

The effects of increasing power in the duel laser configuration ispresented in Fig. 6. In (a) we see the control tissue, in this case, graymatter in the brain with no nanoshells present and heated at5000 W/m2. The inherent absorption is low, at 26 m�1 and thescattering is also relatively low, at 5700 m�1. With these parame-ters, it takes 17.7 min to reach 55 �C, and the DT is low at 1.65 �Cgiving a highly even temperature distribution. This combinationwould be ideal except for the fact that the longer heating time alsoresults in a high temperature in the surroundings. Increasing thepower density, shown in (b), mitigates this issue. Raising thepower density to 20,000 W/m2 lowers the required heating timeto 3.22 min and increases the maximum DT to 4.14 �C. The temper-ature peak is slightly inside the medium on both ends due to therelatively low scattering of 5700 m�3. This causes the small dispar-ity between the entry node where the laser light enters and thepoint of maximum temperature. Also, with this combination, thetemperature of the surrounding tissue drops off quickly and re-mains relatively low. This favorable profile is achieved withoutthe aid of nanoshells.

Fig. 7 looks at a case very similar, except it follows the effect ofthe increase of power in a tissue with inherently higher scattering.Here, human liver tissue with a scattering coefficient of 41,100 m�1

and absorption coefficient of 320 m�1 with no nanoshells is heatedwith dual 5000 W/m2 lasers, then dual 15,000 W/m2 lasers. In thefirst case, shown in (a), an intensity of 5000 W/m2 yields a requiredtime of 17.8 min and a max DT of 1.22 �C. Increasing the intensityto 15,000 W/m2 in liver, shown in (b), has the same effect as it hadin the gray matter in Fig. 6b, except we can see that the peaks are

7 � 1014 nanoshells/m3 density: (a) control tissue exposed to 5000 W/m2 laser, (b)

a b

Fig. 6. Effect of increased power on induced hyperthermia for brain tissue with no nanoshell addition during use of duel lasers: (a) control tissue exposed to two opposing5000 W/m2 lasers, (b) identical tissue exposed to two opposing 20,000 W/m2 lasers.

a b

Fig. 7. Effect of increased power on induced hyperthermia for human liver tissue with no nanoshell addition during use of duel lasers: (a) control tissue exposed to twoopposing 5000 W/m2 lasers, (b) identical tissue exposed to two opposing 15,000 W/m2 lasers.

a b

Fig. 5. Effect of use of duel lasers on induced hyperthermia for subcutaneous fatty tissue with 7 � 1014 nanoshells/m3 density at 15,000 W/m2: (a) control tissue exposed tosingle 15,000 W/m2 laser, (b) identical tissue exposed to two opposing 15,000 W/m2 lasers.

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a b

Fig. 8. Effect of increased nanoshell concentration on induced hyperthermia for human liver tissue at during use of duel 5000 W/m2 lasers: (a) control tissue with nonanoshells, (b) identical tissue with 7 � 1016 nanoshells.

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no longer shifted to the center, and the profile is not as smooth. Theheating time is lowered to 4.23 min and the DT is 4.21 �C with thelowest temperature at the center.

Finally, the effect of nanoshell addition in a two-beam casewithout changing laser power is presented. For comparative pur-poses, human brain gray matter tissue is used again. In Fig. 8a,we see the same as Fig. 6a, gray matter with no nanoshells sub-ject to intensive laser radiation of 5000 W/m2 from opposingsides. Fig. 8b shows the effect of adding 7 � 1016 nanoshells/m3 to the system. The results appear very similar to that inFig. 7b: the required heating time is lowered to 4.33 min andthe DT in the region of interest is moderately low at 3.41 �C.Also, the drop off of temperature in the surrounding tissue is al-most identical.

4. Summary and conclusion

The effects of gold nanoshell distribution density, laser powerintensity, and laser arrangement on induced hyperthermia for fivedifferent tissue types have been presented. It is assumed that theaddition a nanoshell array to a tissue will create additional absorp-tion or scattering that will supplement that inherent for a tissue ata given wavelength. Calculations were performed modeling thepenetration of a 633-nm collimated light source using the P1

approximation to obtain a spectral radiative heat flux profile foreach case, assuming the tissue as an absorbing and isotropicallyscattering, one-dimensional plane parallel medium with constantradiative properties. An FDTD simulation utilizing these resultswas made to simulate the conductive propagation of heat througheach tissue to generate a time-dependent temperature profile. Thetissues analyzed were breast tissue, skin dermis, subcutaneous fat,liver, and gray matter tissue. For each, four nanoshell distributiondensities, each at four power intensities, were tested where theirradiation process was simulated using one then two lasers. Heat-ing commenced in each case until a stopping criteria of 55 �C in theentry node was reached. From the resultant temperature profilesobtained, patterns were observed showing that despite subtle dif-ferences in conductivity, specific heat, and density for all tissues,similar effects from the addition of nanoshells, the addition of laserintensity, or the modification of the laser arrangement – in partic-ular, using two opposing lasers – could be seen, and a few exam-ples could be chosen as representative samples to illustrate these

effects. It can be seen that adding nanoshells or increasing the den-sity of nanoshells in a sample significantly decreases the time re-quired for heating. This is beneficial in cases were a small area istargeted for heating and it is desired to leave the surrounding re-gion unaffected. However, this is not advantageous if a wider areais desired to be heated. The addition of nanoshells to a slab affec-tively increases the absorption and lowers the skin depth of thesample. As a result, the entry region of the slab where the laser firstpenetrates to absorbs most of the incoming radiation while leavinglittle or none to reach the deeper levels, and volumetric heatingonly occurs at the entry boundary leaving the rest of the tissueheated by thermal conduction. In addition, because of rapid heat-ing at the entry, a highly uneven temperature profile develops be-cause of less time for the heat to propagate before the entryreaches the established stopping criteria.

It can be seen that the addition of laser intensity (power den-sity) has a similar effect. With an increased intensity, volumetricheating due to optical absorption in the entry region causes thetemperature in this region to rise rapidly, allowing little time forheat to diffuse evenly into the tissue depth. Without the stoppingcriteria at 55 �C, temperatures at the entry would continue to esca-late, and the entry region would undergo pyrolysis and ablationbefore the rest of the tissue could warm.

The use of two opposing lasers best suits the needs of evenheating across a given depth. This can be achieved with or withoutthe use of gold nanoshells. Rapid heating can be trigged by eitherincreasing the power or increasing the nanoshell concentration.In such a configuration, rapid heating is beneficial, since little timeis allowed for heat to spread to the surroundings, creating a pla-teau-like temperature profile with the highest temperatures pro-duced in the target region. This is best achieved with tissuesexhibiting lower scattering.

This study made several simplifying assumptions, includingconstant temperature thermal properties and the omission ofblood-flow related effects. Additionally, for cancer therapy, tumor-ous tissues have been known to host different optical propertiesthan those in their healthy state. Future studies incorporating suchtissue data and blood flow effects as presented by the bio-heatequation are needed to increase accuracy. For the purposes of pre-senting the general effects of nanoshell addition, laser intensity,and laser configuration on irradiated samples of human tissue,however, this simplified model serves as a good starting point.

J. Vera, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 52 (2009) 564–573 573

Acknowledgements

This work was funded by the Alliances for Graduate Educationand the Professoriate (AGEP) program through the National Sci-ence Foundation Grant HRD-0450363. The kind guidance andassistance from Indra Tjahjono, Joseph Cole and the fruitful semi-nar series from the Laboratory for Nanophotonics (LANP) at Riceis also gratefully acknowledged.

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